Question

Two APs have the same common difference. The difference between their ${100}^{th}$ terms is $100$, what is the difference between their ${1000}^{th}$ terms.

Answer 1

Given that their $100$th term's difference is $100$

Let the first no. of first series be $a_1$ and second series be $a_2.$

then, $a_{100}\left(1\right)-a_{100}\left(2\right)=100$ --- 1)

For $1$st series ---- $a_{100}=a_1+99d$

$2$nd series ---- $a_{100}=a_2+99d$

Put these values in (1)

then,

$a_1+99d-(a_2+99d)=100$

$a_1+99d-a_2-99d=100$

therefore, $a_1-a_2=100$ ---2)

Then, the difference between their $1000$th terms is

for $1$st series --- $a_{1000}=a_1+999d$

for $2$nd series --- $a_{1000}=a_2+999d$

their $100$th terms difference is

$a_{1000}\left(1\right)-a_{1000}\left(2\right)$

$a_1+999d-(a_2+999d)$

$a_1+999d-a_2-999d$

Therefore, we get the value $a_1-a_2$

from (2), $a_1-a_2=100$

Therefore, the difference between their $1000$th terms is $100.$